In this part we will design a simple component analyzer. Use the sound file [https://drive.google.com/uc?export=download&id=1fyHhUqYVOrIPzSMJoQC78sqeokjqI4ib]. This is a drum loop with four distinct sounds (bass drum, snare drum, cymbal and synthetic bell sound). We will use a spectral factorization that will allow us to extract them all. Obtain the STFT of this signal and use a DFT size of 4096, a hop size of 256 and a Hann window. This will be stored in a matrix $\mathbf F$ whose size will be $M$ by $N$.
You now need to implement a factorization technique. This is defined as:
$$|\mathbf F | \approx \mathbf{W} \cdot \mathbf{H}$$$$\mathbf{F} \in \mathbb{R}^{M\times N}_+, \mathbf{W} \in \mathbb{R}^{M\times K}_+, \mathbf{H} \in \mathbb{R}^{K\times N}_+$$
Where $\mathbb{R}^{A\times B}_+$ is the set of matrices of size $A \times B$ containing non-negative elements, and $|\mathbf{F}|$ takes the absolute value of the STFT matrix $\mathbf{F}$. In this case we will use $K$=4 since the mix we are analyzing has four distinct sounds. To estimate the values of $\mathbf{W}$ and $\mathbf{H}$ start by filling them with uniformly random values between 10 to 11 and iterate over the following equations:
$$\mathbf{V} = \frac{|\mathbf{F}|}{\mathbf{W}\cdot \mathbf{H} + \epsilon}$$$$\mathbf{H} = \mathbf{H} \odot \left[ \mathbf{W}^\top \cdot \mathbf{V} \right]$$$$\mathbf{W} = \mathbf{W} \odot \left[ \mathbf{V} \cdot \mathbf{H}^\top \right]$$
Where $\odot$ denotes element-wise multiplication and the fraction performs element-wise division. The constant $\epsilon$ is assigned to a small value (e.g. 1e-7) to avoid division by zero. After each pass normalize the columns of $\mathbf{W}$ to sum to 1. Iterate for about 100 times.
Plot the columns of $\mathbf{W}$ and explain what they correspond to. Plot the rows of $\mathbf{H}$ and explain them as well. You might have to run the above procedure a couple of times since in some cases the results can come up wrong. Just to be safe, run this a dozen times and show the results that are representative of the majority of the outputs (note that each time the ordering will be different, we only care about the shapes of these quantities, not their order).
You can now try to extract each component. Take each column of $\mathbf{W}$ and compute its outer product with its corresponding row of $\mathbf{H}$. This will approximate only one component of the input spectrogram. Plot all four products and explain what they look like. Use the phase of the original input to invert these resulting spectrograms to the time domain and listen to them. What do they sound like?
